RL of building top from stations A and B

A and B are on a line with the building. B sees the rod at A at an angle of elevation, so A is higher and nearer the building. All angles are angles of elevation; AB = 130.570 m is horizontal.

RL of instrument axes

• Axis of B = RL of B + HI = 1234.250 + 1.41 = 1235.660 m
• Sight B to A: elevation 14°11'. The top of the rod (2 m above its foot at A) is at 1235.660 + 130.57 tan(14°11') = 1268.659 m, so the ground at A = 1266.659 m and axis of A = 1266.659 + 1.54 = 1268.199 m.

Let T be the top of the building. Elevation angles to T: from B = 19°28'00", from A = 35°21'00". Take D = 130.57 m as the horizontal distance BA.

Distance and RL of T

Both stations lie in the same vertical plane as T, with A between B and T. Let $x$ = horizontal distance from B to T, so the distance from A to T is $x - D$ with $D = 130.57$ m.

                                  T
                            _.--'|
                      _.--'      |
            A  _.--'  aA        |
   B _.--'  aB  - - - - - - - - |
   |<--- D --->|<--- (x - D) --->|
   |<------------ x ------------>|

Height of T above axis of B = $x\tan\alpha_B$ and above axis of A = $(x-D)\tan\alpha_A$. The RL of T is the same from both stations:

$$\begin{aligned} RL_{axis\,B} + x\tan\alpha_B &= RL_{axis\,A} + (x-D)\tan\alpha_A \\ x &= \frac{(RL_{axis\,A} - RL_{axis\,B}) - D\tan\alpha_A}{\tan\alpha_B - \tan\alpha_A} \end{aligned}$$

With $\alpha_B = 19°28'00"$ ($\tan = 0.35346$), $\alpha_A = 35°21'00"$ ($\tan = 0.70935$), axis RL of B = 1235.660 m and of A = 1268.199 m:

$$x = \frac{(1268.199 - 1235.660) - 130.57\times 0.70935}{0.35346 - 0.70935} = 168.821\ \text{m}$$ $$RL_{T} = 1235.660 + 168.821\times 0.35346 = 1295.332\ \text{m}$$

Check from A: $1268.199 + (168.821 - 130.57)\times 0.70935 = 1295.332$ m (same).

Answer: RL of the top of the building (T) = 1295.332 m.